Probability
Independent Events
Grade 12
Question:
<p>An electrical system has open-closed switches \(S_1\), \(S_2\) and \(S_3\) as shown. The switches operate independently of one another and the current will flow from \(A\) to \(B\) either if \(S_1\) is closed or if both \(S_2\) and \(S_3\) are closed. If \(P(S_1) = P(S_2) = P(S_3) = \frac{1}{2}\), then find the probability that the circuit will work.</p>
Step-by-Step Solution
Key Concept: The circuit works if S₁ is closed OR both S₂ AND S₃ are closed; use P(A∪B) = P(A) + P(B) - P(A∩B) where A is 'S₁ closed' and B is 'both S₂ and S₃ closed', accounting for the fact that these events can overlap.
<p><strong>Step 1:</strong> Identify the two ways current flows:</p><ul><li>Event A: S₁ is closed → P(A) = 1/2</li><li>Event B: Both S₂ AND S₃ are closed → P(B) = P(S₂) × P(S₃) = 1/2 × 1/2 = 1/4</li></ul><p><strong>Step 2:</strong> Circuit works if A OR B occurs. These events are <strong>not mutually exclusive</strong> (both can happen), so:</p><p>P(A ∪ B) = P(A) + P(B) − P(A ∩ B)</p><p><strong>Step 3:</strong> Calculate the overlap:</p><p>P(A ∩ B) = P(S₁ closed AND S₂ closed AND S₃ closed) = 1/2 × 1/2 × 1/2 = 1/8</p><p><strong>Step 4:</strong> Substitute into the formula:</p><p>P(circuit works) = 1/2 + 1/4 − 1/8</p><p>= 4/8 + 2/8 − 1/8 = 5/8</p><p><strong>∴ Answer: 5/8</strong></p>
Correct Answer: 5/8